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EC201 Intermediate Macroeconomics
EC201 Intermediate Macroeconomics

Lecture 29-30: Exogenous Economic Growth: the
Solow Growth Model
Lecture Outline:
- The Solow growth model;
- The Golden Rule;
- Convergence;
Essential reading:
Mankiw: Ch
...
1, 8
...
8
Introduction
Economic growth refers to the study of what determines the growth of output per
capita in the long run
...
In the first part of this module we mainly focused on the short run
fluctuations of output (deviations from the long term trend)
...


1

Economic growth tries to explain what determines shifts in the vertical aggregate
supply
...
In RBC theory shifts in the vertical aggregate
supply were caused by change in the technological progress
...

In particular we are going to study two kinds of growth models
...
In this lecture note we focus on exogenous growth
theory
...

Why is economic growth important? Growth in long run output affects the standard of
living of people in an economy
...
The living standard in a
given economy is normally measured by the real GDP per capita, higher is the real
GDP per capita and higher is the standard of living
...

100

Madagascar

% of population
living on $2 per day or less

90

India
Nepal
Bangladesh

80
70
60

Botswana

Kenya

50

China

40

Peru

30

Mexico

Thailand

20
Brazil

10
0
$0

Chile
Russian
Federation

$5,000

$10,000

S
...
For example by looking at differences
across countries we can see that in 2000 the real GDP per capita was around $34000
(valued at 2000 prices) while in Mexico was $9000 and in Nigeria $1000
...
Difference across time is for

2

example: a boy born in Japan in 1880 had a life expectancy of 35 years and today is
81 years
...
An important question in economic growth is: How
can countries with low level of real GDP per capita catch up with the high levels
enjoyed by the United States or the Western European countries? The answer is only
by high growth rates of real GDP per capita sustained for long periods of time
...

For example, between 1870 and 2000 the US real GDP per capita grew at an average
rate of 1
...
If US had grown at 0
...
75% average growth rate
...
75%, the real GDP per capita in 2000 would be more than twice
the value with a 1
...

An interesting quote from Robert Lucas (Nobel Prize in 1995)1 about the study of
economic growth:
“… Is there some action a government of India could take that would lead the Indian
economy to grow like Indonesia's or Egypt's? If so, what exactly? If not, what is it
about the "nature of India" that makes it so? The consequences for human welfare
involved in questions like these are simply staggering: once one starts to think about
them, it is hard to think about anything else”
Some Stylised Facts about Economic Growth
Before developing a theory of economic growth we start with some stylised facts
obtained from the data
...
Those facts found in 1957 were found
to be true for other countries as well in different periods of time:
Production and labour productivity have increased at a more or less constant growth
rate in the long run;
(a) Output per worker and capital per worker grow roughly at the same rate;
(b) Capital-output ratio has been stable (constant) in the long run;
(c) The return to capital is roughly constant while the wage rate grows at the same
rate as output;
(d) The income shares of labour and capital is roughly constant;
(e) There are differences in the rates of growth of output and labour productivity
among countries;

1

Robert J
...
22
...
In the long run real GDP is produced according a
production function that depends on productive inputs (capital and labour) and
technological progress or “knowledge”
...
The term AL is also called as effective labour
...


Notice: here the technological progress has the same effect of an increase in labour
...

Examples of technological progress:
(a) Percentage of US households with at least one computer was 8% in 1984
while 62% in 2003;
(b) In US in 1981 there were 213 computers connected to the Internet while in
2000 there were 60 million computers connected to the Internet;
(c) In 2001 the iPod capacity was 5 giga byte while in 2006 it was 80gb;

Main properties of the production function:
(a) The production function has Constant Returns to Scale
...
This means that if
we increase both inputs by the same amount λ > 0 also output increases by the
same amount:

F (λK , λAL) = λF ( K , AL) = λY
For example, a Cobb-Douglas production function like F ( K , AL) = K α ( AL)1−α
satisfies the constant return to scale property
...
This is different
from (a)
...
This assumption says that given the
number of workers, as we increase the level of capital, the marginal productivity
of capital decreases
...
This
∆L
means that
= n
...

A
We can think at the rate n as the growth rate of population so labour is growing as
population grows
...
This means that

proportional to output per capita
...
This will give us how

output per effective worker (this is slightly different than output per worker!) is
produced
...
Equation 2) then can be written as:
y = f (k )
3)
where we use f for the production function because we have now a production of
output per effective worker
...
We assume that f (0) = 0
...
This is quite intuitive
...
The production function in 3) looks like the one depicted below
...
Given the level of effective labour, as capital increases, output increases but
by less and less because extra units of capital become less and less productive
...
Equation 4) determines the macroeconomic equilibrium in our
economy
...
Now we introduce an important assumption of the Solow
model
...
In each period a constant fraction of output y is
saved for investment purposes
...

Therefore the equilibrium condition 4) can be written as:
sf (k ) = i
5)
From the national identity we have that consumption in each period can be also
written as:
c = (1 − s ) f (k )

6)

Graphically:
Output per effective

f(k)

worker, y

c
sf(k)

y

i

k

Capital per effective
worker, k

In each period, given the level of capital k we can produce an amount y Of that

6

amount a proportion s is saved and thus invested, a proportion 1 − s is consumed
...
Capital level in any period is a stock, meaning
a variable that moves slowly over time
...
The
investment represents the flow variable that affects the stock of capital
...
Therefore δ k denotes the
fraction of capital in a given period that wears out
...

The term it is simply the investment per effective worker at time t
...


Explanation of break-even investment: there are two reasons why some investment is
needed to keep capital per effective labour from falling
...
This is captured by the term δ k t
...
In particular At Lt grows approximately at rate n + g
...

This is captured by the term (n + g )k t
...
This means that we have 100
effective workers at time t in the economy ( At Lt = 100 )
...
Then capital per effective workers at time t is 1
( kt =

Kt
100
=
= 1 )
...
At time t+1 labour
At Lt 100

and technological progress will be different since they grow
...
At time t+1 we have 2% more workers so Lt +1 = 10
...
Similarly
At +1 = 10
...
In order for capital per
effective worker to remain constant we must have K t +1 = 104
...
Therefore capital must
increase at the same rate for k to remain constant
...
The true growth rate of effective labour is 4
...


7

4
= 0
...

100
= 0
...
02 + 0
...
So

must be equal to 4
...

From equation 7) we have:
(a) ∆k t +1 > 0 ⇒ it > (n + g + δ )k t
...
Capital decreases between

t

and

t+1
...
Capital is constant between t and t+1
...

∆k t +1 = sf (k t ) − (n + g + δ )k t
8)
Definition: the level of capital per effective worker k reaches the steady-state (= the
long-run equilibrium denoted with k * ) when it remains constant over time
...
In that case the capital per effective

workers remains constant and so we are at the steady state
...

We can see the steady state of capital graphically:

8

Investment, break-even
investment

(δ +n +g ) k

sf(k)

Investment

∆k
(n+g+δ)k
0

k*

Capital

per

effective

worker, k

In the graph we have two lines: sf (k t ) denotes investment and it simply the
production function multiplied by a constant ( s )
...
The steady state level of capital is where the two lines intersect
...
This is a trivial equilibrium
...
If there is no production there is no consumption and no investment
and so capital stays at zero
...
Suppose that we are at k = 0 , if nothing happens we stay there
...

b) k = k *
...
It implies a positive level of capital per
effective worker in steady state
...
If we start at
point different from k * the economy converges towards it
...
At k 0 we have

sf (k 0 ) > (n + g + δ )k 0 and so capital per effective worker increases to k1
...
This process stops
when we arrive at k * where sf (k * ) = (n + g + δ )k * and capital remain constant at k *
...
In that case the initial capital is too high and it will decrease towards k *
...

The Balanced Growth Path in the Solow Model
We want to know how the main variables of the model behave when k = k * in terms
of their growth rates at the steady state
...
In equilibrium capital per effective
k*
worker is constant at k * and therefore its steady state growth rate is zero
...
At the steady state this is given by ALk *
...

∆y
= 0
...

∆ (Y / L )
Y
d) Output per worker:
= g
...
We know that the growth rate of y is zero while A grows at rate g ,
L
therefore Ay grows at rate g + 0 = g
...
This is because Y = yAL and yAL grows at rate
Y

n+g
...
At the steady state the variables of the model all grow at some constant
rate (meaning a rate that does not change over time)
...
The most important variable is output per worker (even if we
have done all the analysis using output per effective worker) that can be thought as
the real GDP per capita in the model
...
Therefore it is the technological progress that creates growth
in living standard
...
Capital accumulation has a level effect but not a
growth effect
...
Suppose an increase in the saving rate from s old to s new
...
Graphically an increase in s shift the curve sf (k ) up and so a higher
steady state will be reached
...
However this will not
affect the growth rate of output per worker that will remain equal to g in the long
run
...
Suppose that at t 0 there is a permanent increase in the saving rate
...
Output per
effective worker increases as well towards a higher steady state and then it remains
constant
...

This is shown in the figures below
...

Is there a “best” steady state?
The best steady state will be the one in which consumption per effective worker is the
highest
...

Higher the consumption and higher should be the welfare
...
If output increases then consumption will increase;

13

b) it reduces (1 − s ) and so it should reduce consumption;
Can we find a saving rate and so a capital level that maximizes the steady state level
of consumption per effective worker?
In steady state consumption is given by: c * = y * − i * , where the asterisk says that the
variables are at the steady state level (where k = k * )
...

Therefore consumption per effective worker at the steady state is given by:
c * = f (k * ) − (n + g + δ )k *
10)
What is the steady state capital level k * that maximises consumption c * ?
dc *
We need to find
(the derivative of consumption with respect to capital) from 10)
dk *
and set it equal to zero
...

dk *
The condition f ' (k * ) = n + g + δ is called the Golden Rule
...
The saving
rate associated with k * gold is the Golden Rule Saving Rate called s gold
...
The term
dk *
n + g + δ is the slope of the line (n + g + δ )k
...

This is shown graphically below
...
In the graph below for example the current steady state
level of capital is larger than the golden state level
...
By decreasing the level of capital in the economy,
consumption per capita will increase
...
A government can use policy to
affect the saving rate in order for the economy to be closer to the golden rule capital
level
...
Any
decrease in the saving rate towards s gold will improve consumption per capita for all
generations
...


14

Investment, break-even
investment

(δ +n +g ) k

f(k)

sf (k )
s gold f (k )
Golden rule
consumption
0

k * gold

k*

Capital

per

effective

worker, k

Example: in US the marginal productivity of capital is estimated to be 0
...
Real GDP did grow at an average rate of
3% recently (before the credit crunch)
...
03
...
With those numbers we have 0
...
07 meaning that we are not
at the golden rule level of capital
...
An increase in the level of capital will increase consumption
...
This means that the saving rate is too low
...

Suppose we start from a situation where are below the golden rule level ( k * < k * gold )3
...
In particular
consumption will converge to a higher equilibrium level
...
This is shown in the following figure
...


15

Effect of an increase of s on c when k * < k * gold
s
snew
sold

c

t0

t

t0

t

Predictions of the Solow Model: Convergence
Some of the predictions of the Solow model are:
a) Higher is the saving rate, higher is the steady state level of capital and higher
is the real output per capita
...
In the graph below we has a scatter plot
between real income per person in 2000 in different countries and the average
investment as a percentage of output in between 1960 and 2000 in those
countries
...


16

b) Countries with higher population growth should have lower capital and so
lower output per capita
...
Empirical evidence seems to be
consistent with this prediction as well
...

Suppose two countries A and B that are equal in every aspect part the fact that
A has lower capital per worker and so lower output per worker than B
...
However both countries should converge to
the same steady state in the long run (they just start at a different point but
they should arrive at the same final equilibrium)
...
This kind of
convergence is called absolute convergence
...
This is
shown in the following figure:

17

Countries that were poorer relative to the US in 1960 (horizontal axis) are still
poorer relative to the US 40 years later (vertical axis)
...
Countries can differ in saving rates and population growth
...
What the Solow model really predicts is conditional
convergence: different countries do not need to converge to the same steady
state but they may converge to different steady states
...
Data for full
sample of countries lend support to the conditional convergence hypothesis

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